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Mirrors > Home > ILE Home > Th. List > addlsub | GIF version |
Description: Left-subtraction: Subtraction of the left summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.) |
Ref | Expression |
---|---|
addlsub.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addlsub.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
addlsub.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
addlsub | ⊢ (𝜑 → ((𝐴 + 𝐵) = 𝐶 ↔ 𝐴 = (𝐶 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5774 | . . 3 ⊢ ((𝐴 + 𝐵) = 𝐶 → ((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵)) | |
2 | addlsub.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | addlsub.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | 2, 3 | pncand 8067 | . . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
5 | eqtr2 2156 | . . . . . 6 ⊢ ((((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵) ∧ ((𝐴 + 𝐵) − 𝐵) = 𝐴) → (𝐶 − 𝐵) = 𝐴) | |
6 | 5 | eqcomd 2143 | . . . . 5 ⊢ ((((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵) ∧ ((𝐴 + 𝐵) − 𝐵) = 𝐴) → 𝐴 = (𝐶 − 𝐵)) |
7 | 6 | a1i 9 | . . . 4 ⊢ (𝜑 → ((((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵) ∧ ((𝐴 + 𝐵) − 𝐵) = 𝐴) → 𝐴 = (𝐶 − 𝐵))) |
8 | 4, 7 | mpan2d 424 | . . 3 ⊢ (𝜑 → (((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵) → 𝐴 = (𝐶 − 𝐵))) |
9 | 1, 8 | syl5 32 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) = 𝐶 → 𝐴 = (𝐶 − 𝐵))) |
10 | oveq1 5774 | . . 3 ⊢ (𝐴 = (𝐶 − 𝐵) → (𝐴 + 𝐵) = ((𝐶 − 𝐵) + 𝐵)) | |
11 | addlsub.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
12 | 11, 3 | npcand 8070 | . . . 4 ⊢ (𝜑 → ((𝐶 − 𝐵) + 𝐵) = 𝐶) |
13 | eqtr 2155 | . . . . 5 ⊢ (((𝐴 + 𝐵) = ((𝐶 − 𝐵) + 𝐵) ∧ ((𝐶 − 𝐵) + 𝐵) = 𝐶) → (𝐴 + 𝐵) = 𝐶) | |
14 | 13 | a1i 9 | . . . 4 ⊢ (𝜑 → (((𝐴 + 𝐵) = ((𝐶 − 𝐵) + 𝐵) ∧ ((𝐶 − 𝐵) + 𝐵) = 𝐶) → (𝐴 + 𝐵) = 𝐶)) |
15 | 12, 14 | mpan2d 424 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) = ((𝐶 − 𝐵) + 𝐵) → (𝐴 + 𝐵) = 𝐶)) |
16 | 10, 15 | syl5 32 | . 2 ⊢ (𝜑 → (𝐴 = (𝐶 − 𝐵) → (𝐴 + 𝐵) = 𝐶)) |
17 | 9, 16 | impbid 128 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) = 𝐶 ↔ 𝐴 = (𝐶 − 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 (class class class)co 5767 ℂcc 7611 + caddc 7616 − cmin 7926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-resscn 7705 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 |
This theorem is referenced by: addrsub 8126 subexsub 8127 nn0ob 11594 oddennn 11894 |
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